Transformations for multivariate statistics

This paper derives transformations for multivariate statistics that eliminate asymptotic skewness, extending the results of Niki and Konishi (1986, Annals of the Institute of Statistical Mathematics 38, 371-383). Within the context of valid Edgeworth expansions for such statistics we first derive the set of equations that such a transformation must satisfy and second propose a local solution that is sufficient up to the desired order. Application of these results yields two useful corollaries. First, it is possible to eliminate the first correction term in an Edgeworth expansion, thereby accelerating convergence to the leading term normal approximation. Second, bootstrapping the transformed statistic can yield the same rate of convergence of the double, or prepivoted, bootstrap of Beran (1988, Journal of the American Statistical Association 83, 687-697), applied to the original statistic, implying a significant computational saving. The analytic results are illustrated by application to the family of exponential models, in which the transformation is seen to depend only upon the properties of the likelihood. The numerical properties are examined within a class of nonlinear regression models (logit, probit, Poisson, and exponential regressions), where the adequacy of the limiting normal and of the bootstrap (utilizing the k-step procedure of Andrews, 2002, Econometrica 70, 119-162) as distributional approximations is assessed

Cyclic A-infinity Algebras and Calabi--Yau Structures in the Analytic Setting

This paper considers $A_\infty$-algebras whose higher products satisfy an analytic bound with respect to a fixed norm. We define a notion of right Calabi--Yau structures on such $A_\infty$-algebras and show that these give rise to cyclic minimal models satisfying the same analytic bound. This strengthens a theorem of Kontsevich--Soibelman, and yields a flexible method for obtaining analytic potentials of Hua-Keller. We apply these results to the endomorphism DGAs of polystable sheaves considered by Toda, for which we construct a family of such right CY structures obtained from analytic germs of holomorphic volume forms on a projective variety. As a result, we can define a canonical cyclic analytic $A_\infty$-structure on the Ext-algebra of a polystable sheaf, which depends only on the analytic-local geometry of its support. This shows that the results of Toda can be extended to the quasi-projective setting, and yields a new method for comparing cyclic $A_\infty$-structures of sheaves on different Calabi--Yau varieties.Comment: 41 pages, comments welcom

Stability over cDV singularities and other complete local rings

We characterise subcategories of semistable modules for noncommutative minimal models of compound Du Val singularities, including the non-isolated case. We find that the stability is controlled by an infinite polyhedral fan that stems from silting theory, and which can be computed from the Dynkin diagram combinatorics of the minimal models found in the work of Iyama--Wemyss. In the isolated case, we moreover find an explicit description of the deformation theory of the stable modules in terms of factors of the endomorphism algebras of 2-term tilting complexes. To obtain these results we generalise a correspondence between 2-term silting theory and stability, which is known to hold for finite dimensional algebras, to the much broader setting of algebras over a complete local Noetherian base ring.Comment: 23 pages, 6 figures, several lemmas concerning the silting fan in section 2 have been clarified, Proposition 3.4 has been strengthened, and many typos have been correcte

Convex-Arc Drawings of Pseudolines

A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar---each crossing is incident to an unbounded face---and simple---each crossing point is the crossing of only two curves. We show that these arrangements can be represented by chords of a circle, by convex polygonal chains with only two bends, or by hyperbolic lines. Simple but non-outerplanar arrangements (non-weak) can be represented by convex polygonal chains or convex smooth curves of linear complexity.Comment: 11 pages, 8 figures. A preliminary announcement of these results was made as a poster at the 21st International Symposium on Graph Drawing, Bordeaux, France, September 2013, and published in Lecture Notes in Computer Science 8242, Springer, 2013, pp. 522--52

Change will come in a barrel’: a tribute to Rudolph Jansen

This is a Tribute article, recognising the scholarly and other contributions by long-time human rights and land reform lawyer, Advocate Rudolph Jansen SC, who died in Limpopo Province, South Africa on 25 November 2017. His passion for social justice was matched by a keen wit and abiding sense of humour, a combination that was reflected in his own twist to an iconic statement by Mao Tse-Tung, sardonically remarking that ‘change will come in a barrel’